Problem 50 - Contemporary Precalculus

Question

Formal definitions of limits at infinity and negative infinity are given. Adapt the discussion in Special Topics \(13.2 .\) A to explain how these definitions are derived from the informal definitions given in this section. Let \(f\) be a function, and let \(L\) be a real number. Then the statement \(\lim _{x \rightarrow-\infty} f(x)=L\) means that for each positive number \(\epsilon,\) there is a negative real number \(n\) (depending on \(\epsilon\) ) with this property: $$ \text { If } x

Step-by-Step Solution

Answer

Based on the step-by-step explanation provided above, answer the following question: Question: Explain how the formal definitions of limits at infinity and negative infinity are derived from the informal definitions. Answer: The formal definitions of limits at infinity and negative infinity are derived from the informal definitions by introducing the concepts of epsilon (\(\epsilon\)) and a threshold (\(n\)). Epsilon represents any positive real number, which measures the closeness between the function's values and the limit. The threshold, a negative real number, specifies the boundary that ensures the function values are within the desired closeness as the input tends to negative infinity. The formal definition encapsulates the behavior of a function as its input tends towards negative infinity by ensuring the required level of closeness is achieved for all possible epsilon values. Similarly, by using a similar process, we can derive and explain the limit of a function at positive infinity as well.

Step 1: Understand the statement of the problem

We are given a function \(f\) and a real number \(L\). The statement \(\lim_{x \rightarrow-\infty} f(x)=L\) says that the limit of the function \(f(x)\) as \(x\) approaches negative infinity is equal to \(L\). This means that as x gets arbitrarily large in the negative direction, the values of the function \(f(x)\) get arbitrarily close to the value \(L\).

Step 2: Explain the meaning of the term "epsilon"

In the definition, epsilon (\(\epsilon\)) is any positive real number. It is used to represent a small quantity to measure the closeness between the values of the function \(f(x)\) and the limit \(L\). When we say "for each positive number \(\epsilon\)", it means we consider all possible values of closeness.

Step 3: Explain the meaning of the term "negative real number n"

The definition mentions a negative real number \(n\). This number depends on the choice of epsilon (\(\epsilon\)) and represents a boundary or threshold in the domain of the function \(f\). If \(x\) is smaller (in the negative direction) than this boundary, then the values of the function \(f(x)\) will be close enough to the limit \(L\). The importance of \(n\) is that it tells us how far to the left (in the negative direction) we need to go for the function values to be within \(\epsilon\) of the limit \(L\).

Step 4: Explain the meaning of the inequality condition

The inequality \(|f(x)-L|<\epsilon\) represents the closeness between the function \(f(x)\) and the limit \(L\). In this inequality, \(|f(x)-L|\) represents the absolute difference between the function value and the limit value. When this difference is less than epsilon (\(\epsilon\)), it means the function value \(f(x)\) is close enough to the limit \(L\). The condition "if \(x

Step 5: Relate the definition to limits at infinity and negative infinity

The given definition is derived from the informal definitions of limits at infinity and negative infinity because it captures the essence of what it means for a function to approach a certain value as the input tends to negative infinity. The formal definition uses the concepts of \(\epsilon\), a measure of closeness, and a threshold \(n\) to ensure that the function values indeed approach the limit for all possible levels of closeness desired. This definition encapsulates the behavior of a function as its input tends towards negative infinity, and by a similar process, we can define and explain the limit of a function at positive infinity as well.

Key Concepts

Limits at Infinity

Limits at infinity deal with the behavior of a function as the input either grows very large or very small. Specifically, it describes how the function behaves as the input value heads towards infinity or negative infinity. The concept of limits at infinity is crucial in calculus because it helps analyze the behavior of functions over large scales, providing insight into their end behavior.

• When we say a function approaches a limit as \(x\) approaches infinity (or negative infinity), it's about predicting how the function value stabilizes.
• A limit at infinity describes that as \(x\) becomes very large (positively or negatively), \(f(x)\) nears a specific value, \(L\).

To determine limits at infinity, one often looks at the dominant terms of a function's expression, since these terms dictate behavior as \(x\) grows. Considering both positive and negative infinity enables us to visualize the asymptotic nature of the function's graph.

Epsilon-Delta Definition

The epsilon-delta definition is a formal way to capture the concept of a limit. It gives a precise criterion for what it means for a function \(f(x)\) to approach a limit \(L\). This concept is essential in understanding rigor in calculus.

• The term \(\epsilon\) represents any positive number, indicating a level of closeness between \(f(x)\) and \(L\).
• \(\delta\) is another small positive number that establishes a range around \(x\).

For a limit to exist according to the epsilon-delta definition, for every \(\epsilon > 0\), there must be a corresponding \(\delta > 0\) such that if the input \(x\) is within \(\delta\) units from a particular point \(c\), then \(f(x)\) is within \(\epsilon\) units from \(L\). This approach completes the bridge between intuitive and formal mathematical language.

Limit of a Function

The limit of a function is a foundational concept in calculus. It examines what value the function \(f(x)\) approaches as \(x\) nears a certain point or goes off to infinity. This idea is pivotal in analyzing continuity, derivatives, and the overall behavior of functions.

• Limits help determine the value functions approach, which may not be apparent at the exact point of interest due to undefined conditions.
• The notation \(\lim_{x \to a} f(x) = L\) indicates that as \(x\) gets arbitrarily close to \(a\), the function value \(f(x)\) approaches \(L\).

In practical terms, calculating limits can often reveal insights into functions' behaviors, approximations, and potential points of discontinuity. It's instrumental in frames like differential and integral calculus.

Formal Definition of Limit

The formal definition of a limit ties together the intuitive concept with rigorous mathematical criteria. This structured approach allows mathematicians to prove features about functions and their behavior at specific points. Key components include:

• A specific limit \(L\) that \(f(x)\) approaches as \(x\) nears some value \(a\).
• An \(\epsilon\) value that defines a range for how close \(f(x)\) should be to \(L\).

By employing small \(\epsilon\) and \(\delta\) values, one sets up a precise environment to see if \(f(x)\) truly gets very close to \(L\) as \(x\) nears \(a\). If for every \(\epsilon > 0\), there’s a \(\delta > 0\), such that whenever \(0 < |x - a| < \delta\), then \(|f(x) - L| < \epsilon\), the limit holds at point \(a\). This establishes a strong groundwork for limits, differentiability, and other advanced calculus concepts.

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Problem 49

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Problem 50

Other exercises in this chapter

Problem 49

Use numerical or graphical means to find the limit, if it exists. If the limit of f as x approaches c does exist, answer this question: Is it equal to \(f(c) ?\

View solution

Problem 49

Find the rule of the derivative of the function \(f .\) [See Example 7 and the remarks following it.] $$f(x)=x^{2}+x$$

View solution

Problem 50

(a) Approximate \(\lim _{x \rightarrow 0}(1+x)^{1 / x}\) to seven decimal places. (Evaluate the function at numbers closer and closer to 0 until successive appr

View solution

Problem 50

Find the rule of the derivative of the function \(f .\) [See Example 7 and the remarks following it.] $$f(x)=x^{2}-x+1$$

View solution

Key Concepts

Limits at Infinity

Limits at infinity deal with the behavior of a function as the input either grows very large or very small. Specifically, it describes how the function behaves as the input value heads towards infinity or negative infinity. The concept of limits at infinity is crucial in calculus because it helps analyze the behavior of functions over large scales, providing insight into their end behavior.

• When we say a function approaches a limit as \(x\) approaches infinity (or negative infinity), it's about predicting how the function value stabilizes.
• A limit at infinity describes that as \(x\) becomes very large (positively or negatively), \(f(x)\) nears a specific value, \(L\).

To determine limits at infinity, one often looks at the dominant terms of a function's expression, since these terms dictate behavior as \(x\) grows. Considering both positive and negative infinity enables us to visualize the asymptotic nature of the function's graph.

Epsilon-Delta Definition

The epsilon-delta definition is a formal way to capture the concept of a limit. It gives a precise criterion for what it means for a function \(f(x)\) to approach a limit \(L\). This concept is essential in understanding rigor in calculus.

• The term \(\epsilon\) represents any positive number, indicating a level of closeness between \(f(x)\) and \(L\).
• \(\delta\) is another small positive number that establishes a range around \(x\).

For a limit to exist according to the epsilon-delta definition, for every \(\epsilon > 0\), there must be a corresponding \(\delta > 0\) such that if the input \(x\) is within \(\delta\) units from a particular point \(c\), then \(f(x)\) is within \(\epsilon\) units from \(L\). This approach completes the bridge between intuitive and formal mathematical language.

Limit of a Function

The limit of a function is a foundational concept in calculus. It examines what value the function \(f(x)\) approaches as \(x\) nears a certain point or goes off to infinity. This idea is pivotal in analyzing continuity, derivatives, and the overall behavior of functions.

• Limits help determine the value functions approach, which may not be apparent at the exact point of interest due to undefined conditions.
• The notation \(\lim_{x \to a} f(x) = L\) indicates that as \(x\) gets arbitrarily close to \(a\), the function value \(f(x)\) approaches \(L\).

In practical terms, calculating limits can often reveal insights into functions' behaviors, approximations, and potential points of discontinuity. It's instrumental in frames like differential and integral calculus.

Formal Definition of Limit

The formal definition of a limit ties together the intuitive concept with rigorous mathematical criteria. This structured approach allows mathematicians to prove features about functions and their behavior at specific points. Key components include:

• A specific limit \(L\) that \(f(x)\) approaches as \(x\) nears some value \(a\).
• An \(\epsilon\) value that defines a range for how close \(f(x)\) should be to \(L\).

By employing small \(\epsilon\) and \(\delta\) values, one sets up a precise environment to see if \(f(x)\) truly gets very close to \(L\) as \(x\) nears \(a\). If for every \(\epsilon > 0\), there’s a \(\delta > 0\), such that whenever \(0 < |x - a| < \delta\), then \(|f(x) - L| < \epsilon\), the limit holds at point \(a\). This establishes a strong groundwork for limits, differentiability, and other advanced calculus concepts.